Happy Hour is a tricky phrase in Illinois. By law you can't even advertise or promote a "Happy Hour" unless you're hoping for a visit from the Illinois Liquor Control Commission, with the strong chance of a fine. (It's happened to friends before, no exaggeration. They were fined for the simple fact that the commision took offense to their fliers.)
The state cracked down on the idea of happy hours back in the days when people didn't think much of downing a couple of martinis and then making a drive home. The current law stipulates (among other things) that any special pricing on liquor has to stay consistent for no less than 12 contiguous hours.
That aside, you can still find places that will have food specials around that time period. (And a side effect of the enormous hassle that's involved in being granted a liquor license in this city, is that there's very few bars without some form of kitchen around these days.)
As far as places to go, I've always enjoyed Fado's (despite the fact that it's a chain) on Clark and Grand. The upstairs will probably be full of soccer fans and European ex-pats watching today's game. They might charge a cover to get upstairs. The food's outstanding for bar fare. (I remember speaking to a British friend who was back in the UK for a few months, he was avoiding the fish & chips there - holding out until he could get home and hit Fado.)
Another option is Celtic Crossing's on Chicago & Clark. It's a hole in the wall Irish Pub with Irish barstaff and a fireplace imported from County Clare. Cheap drinks, interesting people, and I'm completely unaware if they have a kitchen or not.
Another option would be Le Bar on Wabash & Chestnut, in the lobby of the Sofitel. Overpriced drinks in a semi-pretentious enviroment, but the people-watching is first rate. If you've washed your hair today and don't have thick black-rimmed glasses, you might not fit in. It's worth hanging out at least once though, just for the crowd.
-Pete